Question

For any positive integer n, define  f_n: (0, ∞) → R as f_n(x) = ∑tan^-1(1/(1 + (x + j)(x + j - 1)) for j ∈ [j = 1, n] for all x ∈ (0, ∞). [Here, the inverse trigonometric function tan^-1x  assumes values in (- π/2, π/2).]

Then, which of the following statement(s) is(are) TRUE?

(A) ∑tan²(f_j(0)) for j ∈ [j=1, 5] = 55

(B) ∑(1 + f_j(0))sec²(f_j(0)) for j ∈ [1, 10] = 10

(C) For any fixed positive integer n, lim_(x →∞) tan(f_n(x)) = 1/n

(D) For any fixed positive integer n, lim_(x →∞) sec²(f_n(x)) = 1

Answer 1

Answer is (D)

Let us check all options as follows:

• Options (A) and (B): It is given that

We know that fn(0) = tan^-1n Therefore, 

 tan²(tan^-1n) = n²

As in this case, x = 0 is not in the given domain. That is, x ∈  (0, ∝).

Hence, options (A) and (B) are incorrect

• Option (C): For any fixed positive integer n, we have

Hence, option (C) is incorrect.

• Option (D):

For any fixed positive integer n, we have

Hence, only option (D) is correct.